On the time growth of the error of the DG method for advective problems

Journal Article · Fri Apr 06 00:00:00 EDT 2018 · IMA Journal of Numerical Analysis

DOI:https://doi.org/10.1093/imanum/dry013· OSTI ID:1609968

Kučera, Václav ^[1]; Shu, Chi-Wang ^[2]

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1. Faculty of Mathematics and Physics, Charles University, Sokolovská, Praha, Czech Republic
2. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

Abstract In this paper we derive a priori $$L^{\infty }(L^{2})$$ and L2(L2) error estimates for a linear advection–reaction equation with inlet and outlet boundary conditions. The goal is to derive error estimates for the discontinuous Galerkin method that do not blow up exponentially with respect to time, unlike the usual case when Gronwall’s inequality is used. While this is possible in special cases, such as divergence-free advection fields, we take a more general approach using exponential scaling of the exact and discrete solutions. Here we use a special scaling function, which corresponds to time taken along individual pathlines of the flow. For advection fields, where the time that massless particles carried by the flow spend inside the spatial domain is uniformly bounded from above by some $$\widehat{T}$$, we derive $$\mathcal{O}$$(hp+1/2) error estimates where the constant factor depends only on $$\widehat{T}$$, but not on the final time T. This can be interpreted as applying Gronwall’s inequality in the error analysis along individual pathlines (Lagrangian setting), instead of physical time (Eulerian setting).

• Research Organization: Brown Univ., Providence, RI (United States)
• Sponsoring Organization: USDOE Office of Science (SC)
• DOE Contract Number: FG02-08ER25863
• OSTI ID: 1609968
• Journal Information: IMA Journal of Numerical Analysis, Vol. 39, Issue 2; ISSN 0272-4979
• Publisher: Oxford University Press/Institute of Mathematics and its Applications
• Country of Publication: United States
• Language: English